1 Answer
1

The derivative in its ordinary sense is local: for any neighborhood of $x$, if $f \equiv g$ on that neighborhood and $f^{(n)}(x)$ exists then $g^{(n)}(x)$ exists and $f^{(n)}(x) = g^{(n)}(x)$. So, equations of the form
$$
x^{(n)}(t) = F(t, x(t), x'(t), \dots, x^{(n-1)}(t))
$$
or
$$
\frac{\partial u}{\partial t}(t, x) = \frac{\partial^2 u}{\partial x^2}(t,x) + F(t, x, u(t, x))
$$
are referred to as local.

Generally, nonlocal denotes that in an equation in question there is something that does not belong to the above category. For instance, we can replace in an ODE the derivative by a derivative of fractional order: the latter does not have the locality property as described in the first paragraph.

Equations of the form
$$
x'(t) = F(t, x(t), x(a(t))),
$$
where $a$ is a given function, are, to the best of my knowledge, (almost) never called nonlocal: the standard name appears to be (functional) (ordinary) differential equations with deviating argument (retarded or delayed if $a(t) < t$, and advanced if $a(t) > t$).

The boundary conditions considered in Byszewski's paper look like "usual" multipoint boundary conditions.

It seems that the author just chose to call problems considered by him functional-differential nonlocal problems. I think it would be proper to ask him directly: ludwik.byszewski@pk.edu.pl.